Nuprl Lemma : increasing_inj

`∀[k,m:ℕ]. ∀[f:ℕk ⟶ ℕm].  Inj(ℕk;ℕm;f) supposing increasing(f;k)`

Proof

Definitions occuring in Statement :  inject: `Inj(A;B;f)` increasing: `increasing(f;k)` int_seg: `{i..j-}` nat: `ℕ` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` function: `x:A ⟶ B[x]` natural_number: `\$n`
Definitions unfolded in proof :  inject: `Inj(A;B;f)` uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` all: `∀x:A. B[x]` implies: `P `` Q` prop: `ℙ` nat: `ℕ` subtype_rel: `A ⊆r B` int_seg: `{i..j-}` guard: `{T}` decidable: `Dec(P)` or: `P ∨ Q` squash: `↓T` false: `False` lelt: `i ≤ j < k` and: `P ∧ Q` le: `A ≤ B` iff: `P `⇐⇒` Q` not: `¬A` rev_implies: `P `` Q` uiff: `uiff(P;Q)` top: `Top` less_than': `less_than'(a;b)` true: `True`
Lemmas referenced :  equal_wf int_seg_wf increasing_wf nat_wf increasing_implies decidable__lt less_than_transitivity1 lelt_wf le_weakening less_than_irreflexivity decidable__int_equal false_wf not-equal-2 not-lt-2 add_functionality_wrt_le add-swap add-commutes le-add-cancel add-associates
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep isect_memberFormation introduction cut lambdaFormation hypothesis extract_by_obid sqequalHypSubstitution isectElimination thin natural_numberEquality setElimination rename hypothesisEquality applyEquality functionExtensionality because_Cache lambdaEquality dependent_functionElimination axiomEquality isect_memberEquality equalityTransitivity equalitySymmetry functionEquality independent_isectElimination unionElimination applyLambdaEquality imageMemberEquality baseClosed productElimination dependent_set_memberEquality independent_pairFormation independent_functionElimination voidElimination addEquality voidEquality intEquality

Latex:
\mforall{}[k,m:\mBbbN{}].  \mforall{}[f:\mBbbN{}k  {}\mrightarrow{}  \mBbbN{}m].    Inj(\mBbbN{}k;\mBbbN{}m;f)  supposing  increasing(f;k)

Date html generated: 2017_04_14-AM-07_33_19
Last ObjectModification: 2017_02_27-PM-03_07_36

Theory : fun_1

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